Thurston boundary: Difference between revisions

In [[mathematics]], the '''Thurston boundary''' of [[Teichmüller space]] of a surface is obtained as the [[Boundary (topology)|boundary]] of its closure in the projective space of functionals on simple closed curves on the surface. ItThe Thurston boundary can be interpreted as the space of projective [[measured foliation]]s on the surface.

Let <math>S</math> be a closed surface. A ''measured foliation'' <math>(\mathcal F, \mu)</math> on <math>S</math> is a [[foliation]] <math>\mathcal F</math> on <math>S</math> which may admit isolated singularities, together with a ''transverse measure'' <math>\mu</math>, i.e. a function which to each arc <math>\alpha</math> transverse to the foliation <math>\mathcal F</math> associates a positive real number <math>\mu(\alpha)</math>. The foliation and the measure must be compatible in the sense that the measure is invariant if the arc is deformed with endpoints staying in the same leaf.{{sfn|Fathi|Laudenbach|PoenaruPoénaru|2012|loc=Exposé 5}}

Two measured foliations are said to be ''equivalent'' if they define the same function on <math>\mathcal S</math> (there is a topological criterion for this equivalence via ''Whitehead moves''). The space <math>\mathcal{PMF}</math> of ''projective measured laminationsfoliations'' is the image of the set of measured laminationsfoliations in the projective space <math>\mathbb P(\mathbb R_+^{\mathcal S})</math> via the embedding <math>i</math>. If the genus <math>g</math> of <math>S</math> is at least 2, the space <math>\mathcal{PMF}</math> is homeomorphic to the <math>6g-7</math>-dimensional sphere (in the case of the torus it is the 2-sphere; there are no measured foliations on the sphere).

Let <math>S</math> be a closed surface. Recall that a point in the Teichmüller space is a pair <math>(X, f)</math> where <math>X</math> is ana hyperbolic surface (a [[Riemannian manifold]] with sectional curvatures all equal to <math>-1</math>) and <math>f</math> a homeomorphism, up to a natural equivalence relation. The Teichmüller space can be realised as a space of functionals on the set <math>\mathcal S</math> of isotopy classes of simple closed curves on <math>\mathcal S</math> as follows. If <math>x = (X, f) \in T(S)</math> and <math>\sigma \in \mathcal S</math> then <math>\ell(x, \sigma)</math> is defined to be the length of the unique closed geodesic on <math>X</math> in the isotopy class <math>f_*\sigma</math>. The map <math>x \mapsto \ell(x, \cdot)</math> is an embedding of <math>T(S)</math> into <math>\mathbb R_+^{\mathcal S}</math>, which can be used to give the Teichmüller space a topology (the right-hand side being given the product topology).

The boundary <math>\overline \mathcal T \setminus \mathcal T</math> is equal to the subset <math>\mathcal{PMF}</math> of <math>\mathbb P(\mathbb R_+^{\mathcal S})</math>. The proof also implies that the Thurston compactfification is homeomorphic to the <math>6g - 6</math>-dimensional closed ball.{{sfn|Fathi|Laudenbach|PoenaruPoénaru|2012|loc=Exposé 8}}

A diffeomorphism <math>S \to S</math> is called [[Pseudo-Anosov map|pseudo-Anosov]] if there exists two transverse measured foliations, such that under its action the underlying foliations are preserved, and the measures are multiplied by a factor <math>\lambda, \lambda^{-1}</math> respectively for some <math>\lambda > 1</math> (called the stretch factor). Using his compactification Thurston proved the following characterisation of pseudo-Anosov mapping classes (i.e. mapping classes which contain a pseudo-Anosov element), which was in essence known to NielseNielsen and is usually called the [[Nielsen-Thurston classification]]. A mapping class <math>\phi</math> is pseudo-Anosov if and only if:

The compactification of Teichmüller space by adding measured foliations is essential in the definition of the [[Ending lamination theorem|ending laminations]] of ana [[hyperbolic 3-manifold]].

A point in Teichmüller space <matH>T(S)</math> can alternatively be seen as a faithful representation of the [[fundamental group]] <math>\pi_1(S)</math> into the isometry group <math>\mathrm{PSL}_2(\mathbb R)</math> of the hyperbolic plane <math>\mathbb H^2</math>, up to conjugation. Such an isometric action gives rise (via the choice of a principal [[ultrafilter]]) to an action on the asymptotic cone of <math>\mathbb H^2</math>, which is a [[real tree]]. Two such actionactions are equivariantly isometric if and only if they come from the same point in Teichmüller space. The space of such actions (endowed with a natural topology) is compact, and hence we get another compactification of Teichmüller space. A theorem of R. Skora states that this compactification is equivariantly homeomorphic the Thurston compactification.<ref>{{ cite book | last = Bestvina | first = Mladen | editor-last1 = | editor-first1 = | title = Handbook of geometric topology | contribution = <math>\mathbb R</math>-trees in topology, geometry and group theory | pages = 55–91 | date = | publisher = North-Holland}}</ref>

*{{cite book | ref=harv | last1=Fathi | first1=Albert | last2=Laudenbach | first2=François | last3=Poénaru | first3=Valentin | title=Thurston's work on surfaces Translated from the 1979 French original by Djun M. Kim and Dan Margalit | series=Mathematical Notes | volume=48 | publisher=Princeton University Press | year=2012 | pages=xvi+254 |ISBN=978-0-691-14735-2}}

*{{cite book | ref=harv | last=Ivanov | first=Nikolai | title=Subgroups of Teichmüller Modular Groups | publisher=American Math. Soc. | year=1992}}